Polyhedral Optimization Lecture 3 – Part 2

Name: Polyhedral Optimization Lecture 3 – Part 2
Uploaded: 2017-09-30T07:18:51+00:00
Duration: PTM46S49
Channel: Damon Parsons
Description: Polyhedral Optimization Lecture 3 – Part 2

Polyhedral Optimization Lecture 3 – Part 2
M. Pawan Kumar Slides available online

Outline Operations on Matroids Maximum Weight Independent Set
Truncation Deletion Contraction Duality of Deletion and Contraction Maximum Weight Independent Set Polytopes

t-Truncation of Matroid
M = (S, I) M’ = (S, I’) X ∈I’ if two conditions are satisfied (i) X ∈I (ii) |X| ≤ t M’ is a matroid Proof?

Example – Uniform Matroid
S = {1, 2, 3, 4, 5, 6} k = 3 t = 2 t-Truncation denoted by M’ Is {1, 2, 3} independent in M’ ? NO

S = {1, 2, 3, 4} k = 3 t = 2 t-Truncation denoted by M’ Is {1, 2} independent in M’ ? YES

S = {1, 2, 3, 4} k = 3 t = 2 t-Truncation denoted by M’ Is {1} independent in M’ ? YES

Example – Linear Matroid
1 2 3 4 2 4 6 8 1 2 3 1 2 2 4 1 2 2 4 t = 3 t-Truncation denoted by M’ Independent in M’ ? NO

1 2 3 4 2 4 6 8 1 2 3 1 2 2 4 1 2 2 4 t = 3 t-Truncation denoted by M’ Independent in M’ ? YES

Example – Graphic Matroid
v0 v1 v4 v2 v5 v3 v6 t = 3 t-Truncation denoted by M’

v0 v1 v4 v2 v5 v3 v6 t = 3 t-Truncation denoted by M’ Independent in M’ ? NO

v0 v1 v4 v2 v5 v3 v6 t = 3 t-Truncation denoted by M’ Independent in M’ ? YES

Deletion M = (S, I) M\s = (S-s, I’)
X ∈I’ if two conditions are satisfied (i) X ⊆ S-s (ii) X ∈I M\s is a matroid Proof?

S = {1, 2, 3, 4, 5, 6} k = 3 s = 2 Matroid after deletion of s denoted by M’ Is {1, 2, 3} independent in M’ ? NO

S = {1, 2, 3, 4, 5, 6} k = 3 s = 2 Matroid after deletion of s denoted by M’ Is {1, 3, 4, 5} independent in M’ ? NO

S = {1, 2, 3, 4, 5, 6} k = 3 s = 2 Matroid after deletion of s denoted by M’ Is {1, 3, 4} independent in M’ ? YES

s 1 2 3 4 2 4 6 8 1 2 3 1 2 2 4 1 2 2 4 Matroid after deletion of s denoted by M’ Independent in M’ ? NO

s 1 2 3 4 2 4 6 8 1 2 3 1 2 2 4 1 2 2 4 Matroid after deletion of s denoted by M’ Independent in M’ ? YES

v0 v1 v4 v2 v5 v3 v6

v0 v1 v4 s v2 v5 v3 v6 Matroid after deletion of s denoted by M’

v0 v1 v4 s v2 v5 v3 v6 Matroid after deletion of s denoted by M’ Independent in M’ ? NO

v0 v1 v4 s v2 v5 v3 v6 Matroid after deletion of s denoted by M’ Independent in M’ ? YES

Effect on Rank Function
M = (S, I) M\s = (S-s, I’) rM\s(X) = rM(X), for all X ⊆ S-s Proof?

Contraction M = (S, I) M/s = (S-s, I’) Assume {s} ∈I
X ∈I’ if two conditions are satisfied (i) X ⊆ S-s (ii) X + s ∈I M/s is a matroid Proof?

Contraction M = (S, I) M/s = (S-s, I’) Assume {s} ∉ I
X ∈I’ if two conditions are satisfied (i) X ⊆ S-s (ii) X ∈I M/s is the same as M\s when {s} ∉ I

S = {1, 2, 3, 4, 5, 6} k = 3 s = 2 Matroid after contraction of s is M’ Is {1, 2, 3} independent in M’ ? NO

S = {1, 2, 3, 4, 5, 6} k = 3 s = 2 Matroid after contraction of s is M’ Is {1, 3, 4} independent in M’ ? NO

S = {1, 2, 3, 4, 5, 6} k = 3 s = 2 Matroid after contraction of s is M’ Is {1, 4} independent in M’ ? YES

s 1 2 3 4 2 4 6 8 1 2 3 1 2 2 4 1 2 2 4 Matroid after contraction of s denoted by M’ Independent in M’ ? NO

s 1 2 3 4 2 4 6 8 1 2 3 1 2 2 4 1 2 2 4 Matroid after contraction of s denoted by M’ Independent in M’ ? YES

v0 v1 v4 v2 v5 v3 v6

v0 v1 v4 s v2 v5 v3 v6 Matroid after contraction of s denoted by M’

v0 v1 v4 s v2 v5 v3 v6 Matroid after contraction of s denoted by M’ Independent in M’ ? NO

v0 v1 v4 s v2 v5 v3 v6 Matroid after contraction of s denoted by M’ Independent in M’ ? YES

Effect on Rank Function
M = (S, I) M/s = (S-s, I’) rM/s(X) = rM(X+s) – rM({s}), for all X ⊆ S-s Proof?

Duality (M/s)* = M*\s Proof?
If {s} ∉ I, deletion doesn’t affect independence If {s} ∉ I, contraction doesn’t affect independence Proof is trivial

Duality (M/s)* = M*\s Proof? Assume {s} ∈I X is independent in (M/s)*
s ∉ X ⟺ X ⊆ B1 for some base B1 of (M/s)* ⟺ X ∩ B2 = Null for some base B2 of M/s ⟺ X ∩ B3 = Null for some base B3 = B2 + s of M ⟺ X ⊆ B4 for some base B4 of M* Hence proved.

Polytopes

Independent Set Matroid M = (S, I) X ⊆ S is independent if X  I
X ⊆ S is dependent if X ∉ I

Weight of an Independent Set
Matroid M = (S, I) Weight function w: S → Non-negative Real Weight of an independent set X The sum of weight of its elements

Weight of an Independent Set
Matroid M = (S, I) Weight function w: S → Non-negative Real Weight of an independent set X w(X) = ∑s∈X w(s)

Example (Uniform Matroid)
s w(s) 1 10 2 5 3 4 6 7 12 8 9 S = {1,2,…,9} k = 4 X = {1, 3, 5}

s w(s) 1 10 2 5 3 4 6 7 12 8 9 S = {1,2,…,9} k = 4 X = {1, 3, 5} w(X)? 15

Example (Graphic Matroid)
v0 S = E 2 6 v1 v4 2 3 1 Forest X 5 v2 v5 3 2 v3 v6 4

v0 S = E 2 6 v1 v4 2 3 1 Forest X 5 v2 v5 3 2 w(X)? v3 v6 4 10

Maximum Weight Independent Set
Matroid M = (S, I) Weight function w: S → Non-negative Real Find an independent set with maximum weight maxX∈I w(X)

Matroid M = (S, I) Weight function w: S → Non-negative Real Find an independent set with maximum weight maxX∈I ∑s∈X w(s)

True or False There exists an optimal solution
that is a base of the matroid TRUE

s w(s) 1 10 2 5 3 4 6 7 12 8 9 S = {1,2,…,9} k = 4 Feasible Solutions?

s w(s) 1 10 2 5 3 4 6 7 12 8 9 S = {1,2,…,9} k = 4 Optimal Solution?

s w(s) 1 10 2 5 3 4 6 7 12 8 9 S = {1,2,…,9} k = 4

v0 S = E 2 6 v1 v4 2 3 1 5 v2 v5 Feasible Solutions? 3 2 v3 v6 4

v0 S = E 2 6 v1 v4 2 3 1 5 v2 v5 Optimal Solution? 3 2 v3 v6 4

v0 S = E 2 6 v1 v4 2 3 1 5 v2 v5 Efficient Algorithm? 3 2 v3 v6 4

Greedy Algorithm Efficiency Optimality: Necessity Optimality: Sufficiency Extensions Polytopes

Greedy Algorithm Start with an empty set Repeat
Pick a new element with maximum weight such that the new set is independent Add the element to the set Until no more elements can be added

Greedy Algorithm X ← ϕ Repeat Pick a new element with maximum weight
such that the new set is independent Add the element to the set Until no more elements can be added

Greedy Algorithm X ← ϕ Repeat s* = argmaxx∈S\X w(s)
such that the new set is independent Add the element to the set Until no more elements can be added

such that X ∪ {s} ∈ I Add the element to the set Until no more elements can be added

such that X ∪ {s} ∈ I X ← X ∪ {s} Until no more elements can be added

s w(s) 1 10 2 5 3 4 6 7 12 8 9 S = {1,2,…,9} k = 4

v0 S = E 2 6 v1 v4 2 3 1 5 v2 v5 3 2 v3 v6 4

Efficiency of Greedy Algorithm
X ← ϕ Repeat s* = argmaxx∈S\X w(s) O(|S|) such that X ∪ {s} ∈ I O(|S|) X ← X ∪ {s} Until no more elements can be added

Independence Testing Oracle
Efficiency depends on independence testing Brute-force is not an option Uniform matroid? Graphic matroid?

Optimality: Necessity
(S, I) is a matroid ⟹ (S, I) admits an optimal greedy algorithm Proof?

Proof Sketch Matroid M = (S, I) Proof by induction Current solution X
X is part of optimal solution B (base of M)

Proof Sketch Next step chooses to add s If s ∈ B, trivial
Otherwise, consider base B’ such that X ∪{s} ⊆ B’ ⊆ B ∪{s} B’ = (B ∪{s})\{t} w(B’) ≥ w(B) by construction

Optimality: Sufficiency
(S, I) is a matroid ⟹ (S, I) admits an optimal greedy algorithm Proof?

Proof Sketch Matroid M = (S, I) X ∈ I Y ∈ I k = |X| < |Y|
There should exist an s ∈ Y\X, X ∪{s} ∈ I Proof by contradiction

Proof Sketch Recall k = |X| k+2, if s ∈ X k+1, if s ∈ Y\X w(s) =
0, otherwise Greedy solution has weight k(k+2) w(Y) ≥ (k+1)(k+1) > k(k+2)

Maximum Weight Base Matroid M = (S, I) Weight function w: S → Real
Find a base with maximum weight maxX∈B w(X) Does greedy work? YES

Minimum Weight Base Matroid M = (S, I) Weight function w: S → Real
Find a base with minimum weight minX∈B w(X) Does greedy work? YES

Matroid M = (S, I) Weight function w: S → Real Find an independent set with maximum weight maxX∈I w(X) Does greedy work? YES

Polytopes

Incidence Vector of Set
Matroid M = (S, I) Set X ⊆ S Incidence vector vX ∈ {0,1}|S| 1, if s ∈ X vX(s) = 0, if s ∉ X

s w(s) 1 10 2 5 3 4 6 7 12 8 9 S = {1,2,…,9} 1 k = 4 X = {1, 3, 5} vX?

v0 S = E e1 e2 v1 v4 e4 e5 e3 X ⊆ S e6 v2 v5 e7 e9 v3 v6 e8

v0 S = E e1 e2 1 v1 v4 e4 e5 e3 X ⊆ S e6 v2 v5 e7 e9 vX? v3 v6 e8

Incidence Vectors of Independent Sets
Matroid M = (S, I) Convex Hull Ax ≤ b vX ∈ {0,1}|S|, X ∈ I A? b? First, a property !! Independent Set Polytope

Matroid M = (S, I) w* = maxX∈I w(X) S = {s1,s2,…,sm}, such that w(si) ≥ w(si+1) ≥ 0 Ui = {s1,…si} Optimal solution Why? X = {si | rM(Ui) > rM(Ui-1)}

Matroid M = (S, I) w* = maxX∈I w(X) w(X) = ∑s∈X w(s) = ∑i w(si)(rM(Ui)-rM(Ui-1)) = ∑i λi rM(Ui) Relationship between w and λ? w = ∑i λi vUi X = {si | rM(Ui) > rM(Ui-1)}

Matroid M = (S, I) w* = maxX∈I w(X) w(X) = w* = ∑i λi rM(Ui) such that w = ∑i λi vUi We need to remember this for what follows

Polytopes Independent Set Polytope Base Polytope Spanning Set Polytope

Independent Set Polytope
Matroid M = (S, I) Convex Hull Ax ≤ b vX ∈ {0,1}|S|, X ∈ I x ∈ Real|S|

Matroid M = (S, I) vX ∈ {0,1}|S|, X ∈ I x ∈ Real|S| Necessary conditions Sufficient for integral x Proof? xs ≥ 0, for all s ∈ S Why? ∑s∈U xs ≤ rM(U), for all U ⊆ S Why?

Matroid M = (S, I) vX ∈ {0,1}|S|, X ∈ I x ∈ Real|S| Necessary conditions Sufficient for all x Proof? xs ≥ 0, for all s ∈ S Why? ∑s∈U xs ≤ rM(U), for all U ⊆ S Why?

miny ∑U⊆S yUrM(U) yU ≥ 0, for all U ⊆ S ∑U⊆S yUvU ≥ w Dual? maxx wTx xs ≥ 0, for all s ∈ S ∑s∈U xs ≤ rM(U), for all U ⊆ S

miny ∑U⊆S yUrM(U) yU ≥ 0, for all U ⊆ S ∑U⊆S yUvU = w Greedy algorithm integral solution maxx wTx x’ xs ≥ 0, for all s ∈ S ∑s∈U xs ≤ rM(U), for all U ⊆ S

miny ∑U⊆S yUrM(U) y’ yU ≥ 0, for all U ⊆ S ∑U⊆S yUvU = w Solution obtained using the property maxx wTx x’ xs ≥ 0, for all s ∈ S ∑s∈U xs ≤ rM(U), for all U ⊆ S

miny ∑U⊆S yUrM(U) y’ yU ≥ 0, for all U ⊆ S ∑U⊆S yUvU = w Relationship between objectives for x’ and y’ ? maxx wTx x’ xs ≥ 0, for all s ∈ S ∑s∈U xs ≤ rM(U), for all U ⊆ S

Dual upper bounds primal x’ and y’ must be optimal solutions ∑s w(s)x’s = ∑U⊆S y’UrM(U) For all integer w, we have integer optimal value Negative values in w? Hence proved

Convex Hull Ax ≤ b Base Polytope Matroid M = (S, I)
vX ∈ {0,1}|S|, X ∈ B x ∈ Real|S|

Base Polytope Matroid M = (S, I) vX ∈ {0,1}|S|, X ∈ B x ∈ Real|S|
∑s∈S xs = rM(S) xs ≥ 0, for all s ∈ S ∑s∈U xs ≤ rM(U), for all U ⊆ S

Convex Hull Ax ≤ b Spanning Set Polytope Matroid M = (S, I)
vX ∈ {0,1}|S|, X ∈ S x ∈ Real|S| S\X? Independent set of M*

Spanning Set Polytope Matroid M = (S, I) vX ∈ {0,1}|S|, X ∈ B
x ∈ Real|S| xs ≥ 0, for all s ∈ S 1-xs ≥ 0, for all s ∈ S ∑s∈U (1-xs) ≤ rM*(U), for all U ⊆ S

x ∈ Real|S| xs ≥ 0, for all s ∈ S xs ≤ 1, for all s ∈ S ∑s∈U (1-xs) ≤ rM*(U), for all U ⊆ S

x ∈ Real|S| xs ≥ 0, for all s ∈ S xs ≤ 1, for all s ∈ S ∑s∈U xs ≥ rM(S) - rM(S\U), for all U ⊆ S

Questions?

Polyhedral Optimization Lecture 3 – Part 2

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